Adaptivity with moving grids

28 C. J. Budd, W. Huang and R. D. Russell
If the function P in (2.26) is taken to be
P =
( M
ξ x
) 2
,
then we obtain MMPDE5 (see Section 3 for an alternative derivation),
∂x
∂t = 1 (
∂
M ∂x )
. (2.28)
τ ∂ξ ∂ξ
This equation can be used to evolve the one-dimensional mesh towards
an equidistributed state. It also has a natural generalization to the PMA
equation derived from the optimally transported meshes we will consider in
Section 3.
2.6.3. Variational methods in higher dimensions
Motivated by (2.27) we can consider a generalization to two dimensions,
which is essentially a form of equidistribution in each coordinate direction
(Huang and Russell 1997b, 1999). This is given by
I(ξ,η) = 1 ∫
[
∇ξ T M −1 ∇ξ + ∇η T M −1 ∇η ] dx dy, (2.29)
2 Ω P
where M is now a symmetric positive definite matrix-valued monitor function,
which is a generalization of the original scalar monitor function. The
Euler–Lagrange equations which define the coordinate transformation at
the steady state are then given by
∇ · (M −1 ∇ξ) =0, ∇ · (M −1 ∇η) =0, (2.30)
where all derivatives are expressed in terms of the physical variables so that
∇ =(∂ x ,∂ y ) T . A moving mesh PDE can then be obtained via the gradient
flow equations, given by
∂ξ
∂t = −P δI
τ δξ , ∂η
∂t = −P δI
τ δη , (2.31)
and we will give more details of this procedure in Section 3. A special case
of this system is given by
M = wI, (2.32)
where w is known as the (scalar) weight function. This corresponds to onedimensional
equidistribution and in the steady state gives the equations
( ) ( )
1 1
∇ ·
w ∇ξ =0, ∇ ·
w ∇η =0. (2.33)
Finding a mesh which satisfies this is called Winslow’s variable diffusion
method (Winslow 1967, 1981).